# How do you use the Intermediate Value Theorem and synthetic division to determine whether or not the following polynomial P(x) = x^4 + 2x^3 + 2x^2 - 5x + 3 have a real zero between the numbers 0 and 1?

Aug 29, 2015

You can't.

#### Explanation:

You can use synthetic division (and the Remainder Theorem) to find

$P \left(0\right) = 3$ (although using division seems like the long way to do this)

and $P \left(1\right) = 3$

If we one of these positive and the other negative, then, since $P \left(x\right)$ is continuous, we could use the Intermediate Value Theorem to conclude that there is a zero in $\left(0 , 1\right)$.

The Intermediate Value Theorem can never be used to conclude that there is not a zero in an interval.

We could conclude that $P \left(x\right)$ does not have exactly one zero between $0$ and $1$. (It must have an even number of zeros in in $\left(0 , 1\right)$.)

To help explain why, here is the graph of a different polynomial function:

graph{ x^4 + 2x^3 + 2x^2 - 5x +1 [-3.067, 3.09, -1.17, 1.91]}

Call this polynomial function $Q \left(x\right)$

Notice that $Q \left(0\right) = 1$ and also $Q \left(1\right) = 1$ but his function does have a zero in $\left(0 , 1\right)$. (It has two of them.)